Reorienting the Logic of Abduction
JOHN WOODS[*]
“It is sometimes said that the highest philosophical gift is to invent important new philosophical problems. If so, Peirce is a major star [in] the firmament of philosophy. By thrusting the notion of abduction to the forefront of philosophers’ consciousness he has created a problem which – I will argue – is the central one in contemporary epistemology.”
Jaakko Hintikka
ABSTRACT: Abduction, still a comparatively neglected kind of premiss-conclusion reasoning, gives rise to the questions I want to consider here. One is whether abduction’s epistemic peculiarities can be readily accommodated in philosophy’s mainline theories of knowledge. The other is whether abduction provides any reason to question the assumption that the goodness of drawing a conclusion from premisses depends on an underlying relation of logical consequence. My answer each time will be no. I will spend most of my time on the first. Much of what I’ll say about the second is a promissory note.
I Introduction
Three facts about today’s logic stand out. Never has it been done with such technical virtuosity. Never has there been so much of it. Never has there been so little consensus about its common subject matters. It would seem that the more we have of it, the less our inclination to get to the bottom of its sprawlingly incompatible provisions. There is nothing remotely like this in real analysis, physical chemistry or population genetics. There is nothing like it in the premiss-conclusion reasonings of politics and everyday life. Left undealt with, one might see in logic’s indifference to its own rivalries some sign of not quite knowing its own mind.
Notwithstanding its rivalrous abundance, there are some matters on which logicians are in wide agreement. One is the idea that the core of logic is the relation of logical consequence. But here too there are multiplicities. While logicians tend to agree on the root importance of logical consequence, they are radically less united on the question of how this relation – or, more accurately, this family of relations – is to be individuated and analyzed. Indeed, there are in modern logic more consequence relations than you can shake a stick at.
Although logic’s dominant focus has been the consequence relation, in the beginning its centrality owed comparatively little to its intrinsic appeal.[1] Consequence was instrumentally interesting; it was thought to be the relation in virtue of which premiss-conclusion reasoning is safe, or whose absence would expose it to risk. Reasoning in turn had an epistemic motivation. Man may be many kinds of animal, but heading the list is his cognitive identity. He is a knowledge-seeking and knowledge-attaining being, to which traits his survival and prosperity are indissolubly linked, and indispensable to which is his capacity to adjust what he believes to what follows from what. We might say then that as long as logic has retained its interest in good and bad reasoning it has retained this same epistemic orientation. Accordingly, a logic of good and bad reasoning carries epistemological presuppositions. Typically, however, they aren’t explicitly developed in the logical literature.[2]
Abduction is a form of premiss-conclusion reasoning, by virtue of a relation that links premises to abduced conclusions in the requisite ways. It would be premature to say that abduction has won a central and well-established place in the research programmes of modern logic, but there are some hopeful signs of progress.[3] In the literature to date there are two main theoretical approaches, each emphasizing the different sides of a product-process distinction. The logical (or product) approach seeks for truth conditions on abductive consequence relations and for such other properties as may be interdefinable with it. The computational (or process) approach constructs computational models of how hypotheses are selected for use in abductive contexts. It is not a strict partition. Between the logical and computational paradigms, abductive logic programming and semantic tableaux abduction occupy a more intermediate position. Whatever its precise details, the gap between logic and computer science is not something I welcome. It distributes the theory of abductive reasoning into different camps that have yet to learn how to talk to one another in a systematic way. A further difficulty is that whereas abduction is now an identifiable research topic in logic - albeit a minority one - it has yet to attain that status in computer science. Such abductive insights as may occur there are largely in the form of obiter dicta attached to the main business at hand.[4] This leaves us awkwardly positioned. The foundational work for a comprehensive account of abductive reasoning still remains to be done.
II Abduction
1. Peirce’s abduction
Although there are stirrings of it in Aristotle’s notion of apagogē,[5] we owe the modern idea of abduction to Peirce. It is encapsulated in the Peircean abduction schema, as follows:
The surprising fact C is observed.
But if A were true, C would be a matter of course.
Hence there is reason to suspect that A is true. (CP, 5.189)
Peirce’s schema raises some obvious questions. How central to abduction is the factor of surprise? How are we to construe the element of suspicion? What we are expected to do with propositions that creep thus into our suspicions? When is an occurrence of something a matter of course? As with many of his better ideas, and deeper insights, Peirce has nothing like a fully developed account of abduction. Even so, the record registers some important insight, seven of which I’ll mention here.
P1. Abduction is triggered by surprise. (CP, 5.189)
P2. Abduction is a form of guessing, underwritten innately by instinct. (Peirce, 1992, p. 128, CP, 5. 171, 7. 220)
P3. A successful abduction provides no grounds for believing the abduced proposition to be true. (Peirce, 1992, p. 178)
P4. Rather than believing them, the proper thing to do with abduced hypotheses is to send them off to experimental trial. (CP, 5. 599, 6. 469-6. 473, 7. 202-219)
P5. The connection between the truth of the abduced hypothesis A and the observed fact C is subjunctive. (CP, 5. 189)
P6. The inference that the abduction licenses is not to the proposition A, but rather that A’s truth is something that might plausibly be suspected. (CP, 5. 189)
P7. The premissory link to Peircean conclusions is non-truth preserving. (CP, 5.
189)
P3 conveys something of particular importance. It is that successful abductions are evidentially inert. They offer no grounds for believing the hypotheses abduced. What, then, is the good of them?
2. Ignorance-problems
Seen Peirce’s way, abductions are responses to ignorance-problems. An agent has an ignorance problem in relation to some epistemic target when it can’t be reached by the cognitive resources presently at his command, or within easy and timely reach of it. If, for some proposition A, you want to know whether A is the case, and you lack the information to answer this question, or to draw it out by implication or projection from what you currently do know, then you have an ignorance-problem with respect to A.
Two of the most common responses to ignorance-problems are (1) subduance and (2) surrender. In the first case, one’s ignorance is removed by new knowledge, and an altered position is arrived at which may serve as a positive basis for new action. In the second case, one’s ignorance is fully preserved, and is so in a way that cannot serve as a positive basis for new action. (New action is action whose decision to perform is lodged in reasons that would have been afforded by that knowledge.) For example, suppose that you’ve forgotten when Barb’s birthday is. If her sister Joan is nearby you can ask her, and then you’ll have got what you wanted to know. This is subduance. On the other hand, if Joan is travelling incognito in Nigeria and no one else is about, you might find that knowing Barb’s birthday no longer interests you. So you might rescind your epistemic target. This would be surrender.
Sometimes a third response is available. It is a response that splits the difference between the prior two. It is abduction. Like surrender, abduction is ignorance-preserving but, like subduance, it offers the agent a positive basis for new action. With subduance, the agent overcomes his ignorance. With surrender, his ignorance overcomes him. With abduction, his ignorance remains, but he is not overcome by it. He has a reasoned basis for new action in the presence of that ignorance. No one should think that the goal of abduction is to maintain that ignorance. The goal is to make the best of the ignorance that one chances to be in.
3. The Gabbay-Woods schema
The nub of abduction can be described informally. There is some state of affairs E that catches your attention. It raises a question you’d like to have an answer to. This is your epistemic target. But you don’t know the answer and aren’t in a position here and now to get one. However, you observe that if some further proposition H were true, then it together with what you already know would enable you to answer the question prompted by E. Then, on the basis of this subjunctive connection, you infer that H is a conjecturable hypothesis and, on that basis, you release it provisionally for subsequent inferential work in the relevant contexts.
More formally, let T be an agent’s epistemic target at a time, and K his knowledge-base at that time. Let K* be an immediate successor of K that lies within the agent’s means to produce in a timely way. Let R be an attainment-relation for T and let ⇝ denote the subjunctive conditional relation. K(H) is the revision of K upon the addition of H. C(H) denotes the conjecture of H and Hc its activation. Accordingly, the general structure of abduction can be captured by what has come to be known as the Gabbay-Woods schema:[6]
1. T! E [The ! operator sets T as an epistemic target
with respect to some state of affairs E]
2. ~(R(K, T) [fact]
3. Subduance is not presently an option [fact]
4. Surrender is not presently an option [fact]
5. H e/ K [fact]
6. H e/ K* [fact]
7. ~R(H, T) [fact]
8. ~R(K(H), T) [fact]
9. H ⇝ R(K (H), T) [fact]
10. H meets further conditions S1, ¼Sn [fact]
11. Therefore, C(H) [sub-conclusion, 1-7]
12. Therefore, Hc [conclusion, 1-8]
It is easy to see that the distinctive epistemic feature of abduction is captured by the schema. It is a given that H is not in the agent’s knowledge-set K. Nor is it in its immediate successor K*. Since H is not in K, then the revision of K by H is not a knowledge-successor set to K. Even so, H ⇝ R(K(H), T) . But that subjunctive fact is evidentially inert with respect to H. So the abduction of H leaves the agent no closer than he was before to arriving at the knowledge he seeks. Though abductively successful, H doesn’t enable the abducer to reach his epistemic target. So we have it that successful abduction is ignorance-preserving.
There are respects in which the G-W schema significantly underdetermines the structure of abduction. Line (10) reflects a major omission. It fails to specify the S-condition for hypothesis-selection. Of course, the devil is in the details. Identifying the Si is perhaps the hardest open problem for abductive logic. In much of the literature it is widely accepted that K-sets must be consistent and that their consistency must be preserved by K(H). This strikes me as unrealistic. Belief sets are often, if not routinely, inconsistent. Also commonly imposed is a minimality condition. There are two inequivalent versions of it. The simplicity version advises that complicated hypotheses should be avoided as much as possible. It is sometimes assumed that truth tends to favour the uncomplicated. I see no reason to accept that. On the other hand, simplicity has a prudential appeal. Simple ideas are more easily understood than complicated ones. But it would be overdoing things to elevate this desideratum to the status of a logically necessary condition. The other version is a form of Quine’s maxim of minimum mutilation. It bids the theorist to revise his present theory in the face of new information in ways that leave as much as possible of the now-old theory intact - its “best bits”, in a manner of speaking. It advises the revisionist to weigh the benefits of admitting the new information against the costs of undoing the theory’s current provisions. This, too, is little more than prudence. No one wants to rule out Planck’s abduction of the quantum, never mind the mangling of old physics that ensued. Another of the standard conditions is that K(H) must entail the proposition for which abductive support has been sought. In some variations inductive implication is substituted. Both I think are too strong. Note also that none of the three – consistency, minimality or implication - could be thought of as process protocols.[7]
The Si are conditions on hypothesis-selection. I have no very clear idea about how this is done, and I cannot but think that my ignorance is widely shared. Small wonder that logicians have wanted to off-load the “logic of discovery” to psychology. I will briefly come back to this later. Meanwhile let’s agree to regard line (10) as a kind of rain check.[8]
4. The yes-but phenomenon
Perhaps it won’t come as much of a surprise to learn of the resistance with which the ignorance-preservation claim has been met when the Gabbay-Woods schema has been presented to (what is by now a sizable number of) philosophical audiences.[9] There are those who think that precisely because it strips good abductions of evidential force, the G-W schema misrepresents Peirce. Others think that precisely because it is faithful to Peirce’s conditions the G-W schema discredits the Peircean concept of abduction. Of particular interest is the hesitation shown by philosophers who are actually inclined to accept the schema, and to accept the Peircean approach. It may be true, they seem to think, that abduction is ignorance-preserving, but it is not a truth to which they take kindly. They would be happier if it weren’t true. Something about it they find unsatisfying. There is a conventional way of giving voice to this kind of reticence. One does it with the words, “Yes, but ¼”. So we may speak of this class of resisters as the ignorance-preservation yes-buts.